Stochastic subgradient method converges at the rate O(k-1/4) on weakly convex functions

We prove that the proximal stochastic subgradient method, applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate O(k-1/4). As a consequence, we resolve an open question on the convergence rate of the proximal stochastic gradient method for minimizing the sum of a smooth nonconvex function and a convex proximable function.